Elasticity and Response in Nearly Isostatic Periodic Lattices

Abstract

The square and kagome lattices with nearest neighbor springs of spring constant $k$ are isostatic with a number of zero-frequency modes that scale with their perimeter. We analytically study the approach to this isostatic limit as the spring constant $k^{\prime }$ for next-nearest-neighbor bonds vanishes. We identify a characteristic frequency ${\omega }^{\ast }\sim \sqrt{k^{\prime }}$ and length $l^{\ast }\sim \sqrt{k/k^{\prime }}$ for both lattices. The shear modulus $C_{44}=k^{\prime }$ of the square lattice vanishes with $k^{\prime }$, but that for the kagome lattice does not.

arXiv comment4 pages, 5 figures, revised text as published in PRL

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